| Model cont. | Model ind. | |
| const | 32.90*** | 34.16*** |
| (0.72) | (0.72) | |
| Y | 0.08*** | |
| (0.02) | ||
| Y = 30 | 0.72 | |
| (0.84) | ||
| Y = 90 | 6.49*** | |
| (1.70) | ||
| N | 174 | 174 |
Matching-group-level clustered standard errors in parentheses.
* p<.1, ** p<.05, ***p<.01
Free-form bargaining experiment
Mia Lu
Martin Stancsics
Introduction
Motivation
- Much work on bargaining in experimental economics (understatement)
- Much less work on free-form bargaining
- especially between more than two players
- The one indispensable player / multiple small players setting has real-world relevance
- Wage bargaining
- An inventor with an idea and multiple investors
- A band, where one member owns the PA system
Research question
- Problem: non-cooperative game theory cannot provide predictions without structure
- E.g. timing of the game, who makes the offers
- NCGT solution is alternating offer games, but a lot depends on minor details (Hart and Mas-Colell 1996)
- How does bargaining power affect bargaining outcomes?
- How well do cooperative game theory solution concepts describe the outcomes?
What we do
- Free-form bargaining between three players
- Almost no structure
- Group-level unrestricted chat
- An interface for proposing and accepting allocations
- No binding decision until the very last second
- Vary the bargaining power of the indispensable player
- How important it is to have all small players on board
- We test, whether:
- Outcomes vary as we would expect, based on bargaining power
- Certain CGT solution concepts provide good predictions
Literature
- Early unstructured bargaining papers (1950s-1990s):
Game and solution concepts
The game
- Players: \(N = \{A, B_1, B_2\}\)
- Value function: \(v: 2^N \to \mathbb{R}\)
- No one can create any value alone: \(v(\{A\}) = v(\{B_i\}) = 0\)
- Player \(A\) is indispensable: \(v(\{B_1, B_2\}) = 0\)
- Small players contribute to the value: \(v(\{A, B_i\}) = Y \in [0, 100]\)
- The more small players the better: \(v(\{A, B_1, B_2\}) = 100\)
How to divide the value between the players?
Shapley value
- Each players gets their average marginal contribution
- Fairness based motivation
- Appealing characterization
- Efficient: whole pie is distributed
- Symmetric: identical (from the pov of \(v\)) players should get the same payoff
- Dummy player property: players contributing nothing get nothing
- Linearity (across games)
- Useful properties
- Always exists
- Always unique
Core
- Excess: what a coalition gets vs what it could achieve on its own
- The set of payoff vectors such that excess is non-negative for all coalitions
- No coalition has an incentive to deviate
- Stability-based concept
- Similar idea to the Nash equilibrium, but for multi-player deviations
- Very intuitive and plausible, but
- Can be empty (existence not guaranteed)
- Can be multi-valued (uniqueness not guaranteed)
Nucleolus
- Maximizes the smallest excess across coalitions
- if the core is non-empty, the nucleolus is in the core
- Mix of stability and fairness
- Useful properties
- Always exists
- Always unique
- Can be thought of as a way to mix the stability-based intuition behind the core with useful properties of the Shapley value
In our game
Experimental design
General structure
- Conducted in the BLU lab (May 2024)
- 4 treatments/sessions: \(Y=10\), \(Y=30\), \(Y=90\), dummy player
- 144 subjects in total (36 subjects per session)
- Timing:
- Instructions with comprehension checks
- Slider task
- Trial round + 5 bargaining rounds (5 minutes each)
- Survey (demographics, reasoning, axioms)
- Average payoff across all rounds + show-up fee
- Conversion: 1 point = CHF 0.6
Role and group assignment
- Player roles are assigned stochastically based on players’ performance in the slider task
- the better they perform, the higher the likelihood of becoming Player A
- Stranger matching for bargaining groups each round
- no set of subjects is matched twice
- Split each session into 6 matching groups (à 6 players) in order to account for dependence between rounds in the analysis
- Bargaining groups are only redrawn within matching groups
Bargaining
- Bargaining via public chat and interface for submitting proposals and current acceptances
- Free-form bargaining:
- Unlimited number of proposals
- No restrictions on the order of proposals and acceptances
- Acceptances are not binding and can be changed any time
- At the end of each bargaining round:
- Current acceptances are taken as final decisions
- A proposal is successful only if all members of a coalition agree on it
Payoffs – aggregate
- Dummy player gets something
- \(Y=10\) and \(Y=30\) treatments look similar
- Player A gets more in \(Y=30\) treatment
- Nucleolus better in terms of ‘shape’ but worse in terms of ‘distance’
Payoff of Player A – regression
Matching-group-level clustered standard errors in parentheses.
* p<.1, ** p<.05, ***p<.01
Payoffs – a deeper look
- In all treatments, equal(ish) split is a frequent outcome
- Even in the dummy player treatment!
- \(Y=10\) and \(Y=30\) treatments still look similar
- In the \(Y=90\) treatment, A can occasionally do much better
Payoff of Player A – non-parametrics
Mann-Whitney test comparing Player A’s payoff across groups
| U statistic | p-value | |
|---|---|---|
| [Y = 10] < [Y = 30] | 13.5 | 0.260189 |
| [Y = 10] < [Y = 90] | 0.0 | 0.002499 |
| [Y = 30] < [Y = 90] | 0.0 | 0.002461 |
- Qualitatively agrees with the nucleolus and disagrees with SV
- Quantitatively closer to SV in \(Y=90\) treatment (cf. plots)
Exploratory observations
Payoffs by coordination outcomes
- In the \(Y=90\) treatment, A only achieves significantly higher than ES by excluding one small player
- Not efficient
- Why doesn’t the excluded player seem to want the remaining 10 points
- The outcomes are still not stable
- excluded player could make a counter-offer
Between matching group variance
Testing the axioms
Efficiency
Symmetry
Dummy player axiom
Stability
Linearity
Linearity
Real-time interactions
Time of proposing and accepting final allocation
- Most final allocations are proposed within the first minute
Time of accepting final allocation
- Most agreement times are well before the end of the bargaining time
- In the \(Y=90\) treatment, partial agreements come later
- Otherwise, agreement times are broadly similar across the main treatments
Chat logs
- Have around 6000 chat messages in total
- Participants interacted intensively over chat
- Quality is rather messy (typos, slang, etc.)
- What to do with it?
- Find words that are relatively more prevalent depending on the outcome
- Classify messages into categories (e.g. fairness-related, small talk, etc.)
- Can also learn things like people’s understanding of the game
- In general, would probably need much more data for proper NLP analysis
Chat – Stability-based reasoning
Chat – Fairness-based reasoning
Chat – Dummy player treatment
Chat – Feedback about experiment
Conclusions
What we did
- Free-form bargaining is important to understand
- More realistic than strictly controlled designs
- NCGT have trouble describing it → check if CGT is predictive
- Players could bargain for 5 minutes without any binding actions
- An interface for proposing and accepting allocations
- Group-level unrestricted chat
- Vary the necessity of including both small players between treatments
Main takeaways
- Lots of equal splits in all treatments
- Nucleolus gives qualitatively correct predictions
- People might think in terms of stability and profitable deviations
- Both fail quantitatively, especially when the big player has lots of bargaining power
- SV seems to be closer in the \(Y=90\) case for non-equal splits
- People’s stated preferences (axiom survey) and actions disagree
Future research
- Chat logs are useful even in lieu of proper analysis
- Going through them reveals a lot about players’ understanding of the game
- We have a rich dataset (real time actions)
- Good for exploratory analysis and to inform study design in the future
- Not sure what aspects to look into more deeply – ideas?
- Differences between matching groups
- Maybe just player heterogeneity
- Or norms / reference values determined in the first couple of rounds?
Thank you
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Appendix
Bargaining interface
Survey (axioms)
